zs

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

受欢迎的三角函数 >

arctan(1/4 x)-pi-arctan(x/(100))=-pi

解答

arctan (\frac{1}{4} x) - π - arctan (\frac{x}{100}) = - π

解答

x = 0

求解步骤

arctan (\frac{1}{4} x) - π - arctan (\frac{x}{100}) = - π

使用三角恒等式改写

arctan (\frac{1}{4} x) - π - arctan (\frac{x}{100})

使用和差化积恒等式:

arctan (s) - arctan (t) = arctan (\frac{s - t}{1 + s t})

= - π + arctan (\frac{\frac{1}{4} x - \frac{x}{100}}{1 + \frac{1}{4} x \frac{x}{100}})

- π + arctan (\frac{\frac{1}{4} x - \frac{x}{100}}{1 + \frac{1}{4} x \frac{x}{100}}) = - π

两边加上

π

- π + arctan (\frac{\frac{1}{4} x - \frac{x}{100}}{1 + \frac{1}{4} x \frac{x}{100}}) + π = - π + π

化简

arctan (\frac{\frac{1}{4} x - \frac{x}{100}}{1 + \frac{1}{4} x \frac{x}{100}}) = 0

使用反三角函数性质

arctan (\frac{\frac{1}{4} x - \frac{x}{100}}{1 + \frac{1}{4} x \frac{x}{100}}) = 0

arctan (x) = a \Rightarrow x = tan (a)

\frac{\frac{1}{4} x - \frac{x}{100}}{1 + \frac{1}{4} x \frac{x}{100}} = tan (0)

tan (0) = 0

tan (0)

使用以下普通恒等式:

tan (0) = 0

tan (0)

tan (x)

周期表（周期为

πn

）：

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

= 0

= 0

\frac{\frac{1}{4} x - \frac{x}{100}}{1 + \frac{1}{4} x \frac{x}{100}} = 0

\frac{\frac{1}{4} x - \frac{x}{100}}{1 + \frac{1}{4} x \frac{x}{100}} = 0

解

\frac{\frac{1}{4} x - \frac{x}{100}}{1 + \frac{1}{4} x \frac{x}{100}} = 0 : x = 0

\frac{\frac{1}{4} x - \frac{x}{100}}{1 + \frac{1}{4} x \frac{x}{100}} = 0

化简

\frac{\frac{1}{4} x - \frac{x}{100}}{1 + \frac{1}{4} x \frac{x}{100}} : \frac{96 x}{400 + x ^{2}}

\frac{\frac{1}{4} x - \frac{x}{100}}{1 + \frac{1}{4} x \frac{x}{100}}

\frac{1}{4} x \frac{x}{100} = \frac{x ^{2}}{400}

\frac{1}{4} x \frac{x}{100}

分式相乘:

a \cdot \frac{b}{c} \cdot \frac{d}{e} = \frac{a \cdot b \cdot d}{c \cdot e}

= \frac{1 \cdot xx}{4 \cdot 100}

1 \cdot xx = x^{2}

1 \cdot xx

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

xx = x^{1 + 1}

= 1 \cdot x^{1 + 1}

整理后得

= x^{2}

= \frac{x ^{2}}{4 \cdot 100}

数字相乘:

4 \cdot 100 = 400

= \frac{x ^{2}}{400}

= \frac{\frac{1}{4} x - \frac{x}{100}}{1 + \frac{x ^{2}}{400}}

化简

\frac{1}{4} x - \frac{x}{100} : \frac{6 x}{25}

\frac{1}{4} x - \frac{x}{100}

乘

\frac{1}{4} x : \frac{x}{4}

\frac{1}{4} x

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot x}{4}

乘以:

1 \cdot x = x

= \frac{x}{4}

= \frac{x}{4} - \frac{x}{100}

4, 100

的最小公倍数:

100

4, 100

最小公倍数 (LCM)

4

质因数分解:

2 \cdot 2

4

4

除以

24 = 2 \cdot 2

= 2 \cdot 2

100

质因数分解:

2 \cdot 2 \cdot 5 \cdot 5

100

100

除以

2100 = 50 \cdot 2

= 2 \cdot 50

50

除以

250 = 25 \cdot 2

= 2 \cdot 2 \cdot 25

25

除以

525 = 5 \cdot 5

= 2 \cdot 2 \cdot 5 \cdot 5

2, 5

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 5 \cdot 5

将每个因子乘以它在

4

或

100

中出现的最多次数

= 2 \cdot 2 \cdot 5 \cdot 5

数字相乘:

2 \cdot 2 \cdot 5 \cdot 5 = 100

= 100

根据最小公倍数调整分式

将每个分子乘以其分母转变为最小公倍数所要乘以的同一数值

100

对于

\frac{x}{4} :

将分母和分子乘以

25

\frac{x}{4} = \frac{x \cdot 25}{4 \cdot 25} = \frac{x \cdot 25}{100}

= \frac{x \cdot 25}{100} - \frac{x}{100}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{x \cdot 25 - x}{100}

同类项相加:

25 x - x = 24 x

= \frac{24 x}{100}

约分:

4

= \frac{6 x}{25}

= \frac{\frac{6 x}{25}}{1 + \frac{x ^{2}}{400}}

使用分式法则:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{6 x}{25 ( 1 + \frac{x ^{2}}{400} )}

化简

1 + \frac{x ^{2}}{400} : \frac{400 + x ^{2}}{400}

1 + \frac{x ^{2}}{400}

将项转换为分式:

1 = \frac{1 \cdot 400}{400}

= \frac{1 \cdot 400}{400} + \frac{x ^{2}}{400}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 \cdot 400 + x ^{2}}{400}

数字相乘:

1 \cdot 400 = 400

= \frac{400 + x ^{2}}{400}

= \frac{6 x}{25 \cdot \frac{x ^{2} + 400}{400}}

乘

25 \cdot \frac{400 + x ^{2}}{400} : \frac{400 + x ^{2}}{16}

25 \cdot \frac{400 + x ^{2}}{400}

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{( 400 + x ^{2} ) \cdot 25}{400}

约分:

25

= \frac{400 + x ^{2}}{16}

= \frac{6 x}{\frac{400 + x ^{2}}{16}}

使用分式法则:

\frac{a}{\frac{b}{c}} = \frac{a \cdot c}{b}

= \frac{6 x \cdot 16}{400 + x ^{2}}

数字相乘:

6 \cdot 16 = 96

= \frac{96 x}{400 + x ^{2}}

\frac{96 x}{400 + x ^{2}} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

96 x = 0

两边除以

96

96 x = 0

两边除以

96

\frac{96 x}{96} = \frac{0}{96}

化简

x = 0

x = 0

x = 0

将解代入原方程进行验证

将它们代入

arctan (\frac{1}{4} x) - π - arctan (\frac{x}{100}) = - π

检验解是否符合

去除与方程不符的解。

检验

0

的解:

真

0

代入

n = 1

0

对于

arctan (\frac{1}{4} x) - π - arctan (\frac{x}{100}) = - π

代入

x = 0

arctan (\frac{1}{4} \cdot 0) - π - arctan (\frac{0}{100}) = - π

整理后得

- 3.14159 \dots = - 3.14159 \dots

\Rightarrow 真

x = 0

作图

流行的例子

(cot(x)+1/(cot(x)))/(cos(x))=sec(x)

\frac{cot ( x ) + \frac{1}{c o t ( x )}}{cos ( x )} = sec (x)

sec(θ)=2,(tan(θ)-sin(θ))^2+(1-cos(θ))^2

sec (θ) = 2, (tan (θ) - sin (θ))^{2} + (1 - cos (θ))^{2}

(1+csc(A))(1-csc(A))=-cot(A)

(1 + csc (A)) (1 - csc (A)) = - cot (A)

6cos^2(3x)-cos(3x)-2=0

6 cos^{2} (3 x) - cos (3 x) - 2 = 0

cos(2x)-3cos(x)+1=0

cos (2 x) - 3 cos (x) + 1 = 0